Debarre–Huybrechts–Macrì–Voisin conjecture on numerical invariants of hyper-Kähler manifolds

Let $X$ be a hyper-Kähler manifold of dimension $2n$ with classes $\mathsf l,\mathsf m\in H^2(X,\mathbb Z)$ such that

$$\int_X\mathsf l^{2n}=0\quad\textnormal{and}\quad\int_X\mathsf l^n\mathsf m^n=n!.$$

Debarre–Huybrechts–Macrì–Voisin conjecture. Then $c_X=(2n-1)!!$ and the Huybrechts–Riemann–Roch polynomial of $X$ is

$$P_{RR,X}(T)=\binom{\frac12 T+1+n}{n}.$$

The conjecture concerns the possible Huybrechts–Riemann–Roch polynomials when the Beauville form represents zero. It is known in dimension four ($n=2$), while the paper studies the dimension-six case and obtains an almost-complete result.